I give a proof of this result -- actually the stronger result that the GCD of any two algebraic integers may be expressed as a linear combination -- in $\S 23.4$ of these commutative algebra notes.

The required input from algebraic number theory is nontrivial -- namely that the ideal class group of (the ring of integers of) a number field is finite -- but is much less than that of class field theory.

Note though that one could get away with knowing that these ideal class groups are torsion abelian groups, which is, a priori, a more structural and thus possibly easier to prove result. I have been racking my brains trying to come up with a more fundamentally commutative algebraic proof of this fact, thus far without success.

I give a proof of this result -- actually the stronger result that the GCD of any two algebraic integers may be expressed as a linear combination -- in $\S 23.4$ of these commutative algebra notes.

The required input from algebraic number theory is nontrivial -- namely that the ideal class group of (the ring of integers of) a number field is finite -- but is much less than that of class field theory.

Note though that one could get away with knowing that these ideal class groups are torsion abelian groups, which is, a priori, a more structural and thus possibly easier to prove result. I have been racking my brains trying to come up with a more fundamentally commutative algebraic proof of this fact, thus far without success.